I came across the Stokes equation expressed in following form:
I am trying to expand to check if it is correct but having hard time evaluating it. Can anyone give some hint on how can i expand it to analyze Stokes equation
Here, $\Bbb{u}_i = (u(x,y,z),v(x,y,z),w(x,y,z)), \mu = viscosity \ constant$
I don't understand why you added the subscript $i$ to the velocity vector $\boldsymbol{u}$ and to the pressure $p$.
Anyway, the equation that you posted above is not a closed problem, you need the mass balance equation (also called incompressibility constrain or continuity equation) to close the problem. That is to say:
\begin{equation} \boldsymbol{\nabla} \cdot \boldsymbol{u}=0 \end{equation}
It is easy to verify that the form that you posted is equivalent (with the exception of the minus sign to the pressure term that you forgot) to the more common form: \begin{equation} -\boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^{2} \boldsymbol{u}=\boldsymbol{f} \end{equation}
indeed it is trivial to see that $-\boldsymbol{\nabla} \cdot p \boldsymbol{I}= -\boldsymbol{\nabla} p$.
Another trivial equivalence is $\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \boldsymbol{u}= \boldsymbol{\nabla}^2 \boldsymbol{u}$
Whereas the term $\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \boldsymbol{u}^{T} = \boldsymbol{0}$ because of the mass balance $\boldsymbol{\nabla}\cdot \boldsymbol{u}=0$ (this equivalence is trivial if you express the velocity gradient transpose and the divergence in indicial notation)